# [ $L_p$ ] ERROR ESTIMATES AND SUPERCONVERGENCE FOR FINITE ELEMENT APPROXIMATIONS FOR NONLINEAR HYPERBOLIC INTEGRO-DIFFERENTIAL PROBLEMS

• Li, Qian (School of Mathematics, Shandong Normal University) ;
• Jian, Jinfeng (School of Mathematics, Shandong Normal University) ;
• Shen, Wanfang (School of Mathematics, Shandong Normal University)
• Published : 2005.06.25

#### Abstract

In this paper we consider finite element methods for nonlinear hyperbolic integro-differential problems defined in ${\Omega}\;{\subset}\;R^d(d\;{\leq}\;4)$. A new initial approximation of $u_t(0)$ is taken. Optimal order error estimates in $L_p$ for $2\;{\leq}\;p\;{\leq}\;{\infty}$ are established for arbitrary order finite element. One order superconvergence in $W^{1,p}$ for $2\;{\leq}\;p\;{\leq}\;{\infty}$ are demonstrated as well.